Archive of Applied Mechanics

, Volume 86, Issue 4, pp 669–686 | Cite as

A damage tolerance analysis for complex structures

  • Daniel Pucknat
  • Robert LiebichEmail author


The assessment of damage is usually performed to determine safety or residual life, which is necessary to define inspection intervals for instance. The concept of a damage tolerance analysis presented in this paper ties on and tries to reveal the possible redundancy that statically indeterminate complex structures provide. Classic strength assessments are not able to map changes in load paths or structural supporting effects and that might lead to a huge waste of potential in weight reduction. The damage tolerance analysis is applied to an exemplary complex structure and shows that the fatigue driven damages are not critical for further operation at the assessed load level. The concept uses a new method for the calculation of equivalent stress—the ‘Modified Mohr Mises’ (MMM)-Hypothesis. The MMM-Hypothesis allows the assessment of non-proportional stresses within a fully automated process, due to its invariant equivalent stress notation. Those non-proportional stresses are most common in complex structures of aircrafts, spacecrafts and vehicles for instance.


Damage tolerance Fatigue Crack propagation MMM-Hypothesis Mises-Hypothesis Non-proportional loading 

List of symbols

\(\varepsilon \)


\(\sigma _{\mathrm{eq},a,\mathrm{MMM}}\)

Equivalent amplitude of MMM stress


Interim invariant radius of Mohr


Macro-supporting effect factor

\(\sigma _1, \sigma _2 , \sigma _3\)

Principal stresses according to MMM

\(\varepsilon _a\)

Strain amplitude

\(\hat{{\sigma }}_{a,1}\)

Highest alternating principal stress

\(\eta \)

Dimensionless time


Normalized stress gradient

\(\hat{{\varepsilon }}_\mathrm{H}\)

Linear elastic maximum (Hooke-) strain

\(\sigma _{\mathrm{loc}}^*\)

Local stress in equivalent tension specimen

Open image in new window

Lower Neuber stress level

\(\sigma _{\mathrm{eq},m}\)

Equivalent MMM mean stress


Iteration load cycle

\(\sigma _{-1,N}\)

Compression–tension fatigue limit for certain load cycle

\(\sigma _{\mathrm{eq},\mathrm{Mises}}\)

Equivalent von Mises stress

\(\sigma _x^o , \sigma _y^o , \sigma _z^o\)

Normal stresses from FEM, BEM calculation

\(\sigma \)



Invariant center of Mohr


Sign function


\(\hbox {K}_{\mathrm{t}}\)\(\hbox {K}_{\mathrm{f}}\) ratio (micro-supporting factor)


Tension–shear endurance limit ratio


Load cycle

\(\hat{{\sigma }}_{o,1}\)

Highest maximum principal stress

\(\varphi \)

Constraint factor

\(\hat{{\sigma }}_\mathrm{H}\)

Linear elastic maximum (Hooke-) stress

\(\sigma _{\mathrm{loc}}\)

Local stress in notch

\(\varepsilon _{\mathrm{loc}}\)

Local strain in notch

Open image in new window

Lower Neuber strain level

\(\sigma _{\mathrm{eq},A,\mathrm{MMM}}\)

Sustainable equivalent amplitude of MMM stress


Endurance load cycle


Tension–shear limit ratio for certain load cycle

\(\sigma ^{\prime }\)

Stresses without supporting effects

\(\tau _{xy}^o, \tau _{xz}^o, \tau _{yz}^o\)

Shear stresses from FEM, BEM calculation


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Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.School of Mechanical Engineering and Transport Systems, Institute of Engineering Design, Micro and Medical Technology, Institute of Aeronautics and Astronautics, Chair of Engineering Design and Product ReliabilityTechnical University of BerlinBerlinGermany

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